Biomedical Engineering Reference
In-Depth Information
To cope with the
(b)
and
(c)
limitations, we replace the quadratic cost by
robust functions. To face the problem
(a)
, we use a multiresolution plan and a
multigrid strategy to improve the minimization at each resolution level.
We have here introduced a simple regularization term that makes almost
no assumption on the estimated deformation field. One could imagine choosing
different regularizations for the different brain tissues, but that involves specific
assumptions on the “real” deformation that we do not address in that paper. How-
ever, the introduction of a robust estimator on the regularization term makes it
possible to take into account possible discontinuities on the border of structures
having different physical properties.
8.3.2.2
Rigid Registration Step
Given two images with potentially large displacement, it first seems reasonable
to estimate a rigid transformation. This step is performed by estimating a rigid
transformation that maximizes mutual information [33, 145]. Given two images
A
and
B
, considered as discrete random variables, let us note
p
A
(
a
) and
p
B
(
b
) their
respective marginal probability distribution, and
p
A
,
B
(
a
,
b
) the joint distribution.
Mutual information
I
(
A
,
B
) is then defined as [33, 145]:
p
A
,
B
(
a
,
b
)
p
A
(
a
)
p
B
(
b
)
=
H
(
A
)
+
H
(
B
)
−
H
(
A
,
B
)
,
I
(
A
,
B
)
=
p
A
,
B
(
a
,
b
) log
2
a
,
b
with
H
(
A
)
=−
p
A
(
a
) log
2
(
p
A
(
a
))
and
a
H
(
A
,
B
)
=−
p
A
,
B
(
a
,
b
) log
2
(
p
A
,
B
(
a
,
b
))
.
a
,
b
In some particular cases, such as brain images for instance, it is possible to
define a reference coordinate system that takes into account some information
about the scene (such as resolution of pixels/voxels, orientation of axes, etc.).
In such cases, the two volumes to be registered are mapped in this reference
coordinate system and the rigid transformation is expressed in this coordinate
system. If this a priori information is not available, the rigid transformation is
estimated in the coordinate system attached to the data.
The registration is performed through a multiresolution optimization scheme
(construction of a pyramid of volumes by successive isotropic Gaussian filtering
